Projected Quantum Feature Maps

• Projected Quantum Feature Maps are hybrid methods that encode classical data into quantum states using a data-dependent unitary and then project these states to extract classical features through fixed measurements.
• The approach scales linearly with the number of samples by using a predetermined set of observables, offering a NISQ-compatible alternative to fidelity-based kernels with quadratic sample cost.
• These models integrate quantum state projections with classical learners in applications like credit-default prediction and machine-failure detection, achieving practical performance improvements.

Projected quantum feature maps (PQFMs) are hybrid quantum-classical representations in which a classical input is encoded into a quantum state by a data-dependent unitary and then mapped back to a finite classical vector through a fixed set of measurements. In the recent literature, this paradigm appears under closely related labels including Projected Quantum Feature Maps, Projected Quantum Kernels, and projected quantum feature models. The common objective is to use a quantum processor as a feature generator whose outputs feed a classical learner, rather than to keep the entire learning pipeline quantum. For a fixed number of observables, the costly quantum stage scales linearly with the number of samples, which is why PQFMs are repeatedly positioned as a NISQ-compatible alternative to fidelity-based quantum kernels with quadratic sample cost (Carmo et al., 11 Jun 2026).

1. Origins, terminology, and formal definition

A related antecedent appears in finite-feature-space quantum kernel methods, where kernel evaluations are outsourced to projective measurements on quantum states encoding the data and model training is carried out classically. In that setting, the data are embedded into a fixed finite Hilbert space and classification proceeds from a measured Gram matrix (Bartkiewicz et al., 2019).

In the modern PQFM formulation, an input vector $x\in\mathbb R^p$ is first encoded into a quantum state through a data-dependent unitary. One standard definition writes

$$|\psi(x)\rangle \equiv U(x)\,|0\rangle^{\otimes n},$$

after which a projected classical feature vector is produced by measuring a fixed collection of observables. In the subsystem-based formulation, one chooses $m$-qubit subsystems $M\in S(n)_m$ and a Pauli set $\mathcal P$, defines the reduced state

$$\rho(x)_M=\operatorname{Tr}_{\neg M}\!\left[|\psi(x)\rangle\langle\psi(x)|\right],$$

and then sets

$$\phi_{\mathrm{proj}}(x)_{[(M,O)]}=\operatorname{Tr}[O\,\rho(x)_M], \qquad O\in\mathcal P.$$

The dimensionality is $|S(n)_m|\cdot|\mathcal P|$; in the six-qubit IoT study, this yielded 18 features for local one-body measurements, 18 for nearest-neighbor two-body measurements, and 36 for their union (d'Amore et al., 20 May 2025).

Several application papers use a one-body specialization. In projected quantum feature models for credit default prediction, the observables are $O_{3j}=X_j$, $O_{3j+1}=Y_j$, and $$|\psi(x)\rangle \equiv U(x)\,|0\rangle^{\otimes n},$$0, so the projected representation has dimension $$|\psi(x)\rangle \equiv U(x)\,|0\rangle^{\otimes n},$$1 and is passed directly to a classical learner (Ferenczi et al., 1 Oct 2025). An equivalent 1-RDM description appears in machine-failure detection: each single-qubit reduced state is expanded as

$$|\psi(x)\rangle \equiv U(x)\,|0\rangle^{\otimes n},$$2

and the concatenated coefficients $$|\psi(x)\rangle \equiv U(x)\,|0\rangle^{\otimes n},$$3 form the classical feature vector (Bowden et al., 22 Jan 2026).

2. Projected kernels and hybrid readouts

Once a projected feature vector is available, two routes are common. The first is direct use in a classical model. The second is to define a classical kernel on the projected features. In projected quantum kernels for IoT data, the measured expectation values are assembled into $$|\psi(x)\rangle \equiv U(x)\,|0\rangle^{\otimes n},$$4 and then used inside a Gaussian kernel,

$$|\psi(x)\rangle \equiv U(x)\,|0\rangle^{\otimes n},$$5

The resulting Gram matrix is then supplied to a classical SVM together with a $$|\psi(x)\rangle \equiv U(x)\,|0\rangle^{\otimes n},$$6-regularization parameter (d'Amore et al., 20 May 2025).

A more general framework is the projected hybrid model based on active quantum subspaces. There the input is split as $$|\psi(x)\rangle \equiv U(x)\,|0\rangle^{\otimes n},$$7, only an information-bearing subset is lifted to a quantum representation, and a small set of observables $$|\psi(x)\rangle \equiv U(x)\,|0\rangle^{\otimes n},$$8 defines

$$|\psi(x)\rangle \equiv U(x)\,|0\rangle^{\otimes n},$$9

With any classical feature map $m$0, the hybrid feature becomes

$m$1

and the corresponding kernel is

$m$2

This immediately implies positive semidefiniteness, and the same work proves the rank bound

$m$3

together with

$m$4

The same analysis also gives a necessary and sufficient criterion for improvement over a purely classical predictor in squared loss: the projected quantum sector must contain a direction outside the classical feature span that correlates with the classical residual (Bang et al., 30 May 2026).

The active-subspace theory further derives a PAC sample-complexity bound in a realizable noisy-oracle model,

$m$5

so sample complexity scales as the inverse square of oracle reliability. For a canonical Clifford active-subspace family under local dephasing, the reliability can remain inverse-polynomial even when the encoding gate complexity grows polynomially with system size (Bang et al., 30 May 2026).

3. Hamiltonian and circuit constructions

The Hamiltonian-based PQFM line begins from the observation that projected features can be generated by short evolutions under data-dependent many-body dynamics. Earlier constructions include one-dimensional Heisenberg chains,

$m$6

and counterdiabatic Ising-glass maps, where local fields encode raw features, couplings encode pairwise statistical correlations such as mutual information, and a counterdiabatic term drives fast nonadiabatic entanglement. Their stated limitations are equally specific: Heisenberg maps are constrained to a 1D chain, encode only one feature per qubit per layer, require measurements along all three axes, and are classically simulable by matrix-product-state methods, while CD-Ising-glass maps restrict interactions to $m$7–$m$8 channels and limit per-qubit information density to the number of encoding layers (Carmo et al., 11 Jun 2026).

The generalized two-qubit Hamiltonian PQFM extends these constructions to arbitrary local Pauli fields and hardware-compatible two-qubit Pauli channels. It is defined by

$m$9

where $M\in S(n)_m$0 selects active axes, $M\in S(n)_m$1 is the interaction graph, and $M\in S(n)_m$2 selects the included two-qubit channels. The state is prepared from a fiducial product state, for example $M\in S(n)_m$3, through

$M\in S(n)_m$4

and the projected features are the one- and two-local expectation values

$M\in S(n)_m$5

Because distinct classical variables can be embedded along different Pauli axes of the same qubit, the information density increases: with $M\in S(n)_m$6 qubits and $M\in S(n)_m$7 active axes, a single layer can encode up to $M\in S(n)_m$8 features, and with $M\in S(n)_m$9 layers the capacity is $\mathcal P$0 (Carmo et al., 11 Jun 2026).

For near-term implementation, the generalized construction uses a first-order Trotter approximation in the fast-driving regime,

$\mathcal P$1

with a single Trotter step. Feature-to-slot assignment can be one-to-one or can reuse a feature across multiple axes, while pairwise statistics such as mutual information can be matched to hardware edges through a small genetic algorithm (Carmo et al., 11 Jun 2026).

Concrete specializations of this Hamiltonian perspective appear in applied work. In credit-default prediction, the quantum feature map simulates a one-dimensional Heisenberg chain with Trotterization,

$\mathcal P$2

with even-odd entangling blocks acting on initially Haar-random single-qubit product states (Ferenczi et al., 1 Oct 2025). In machine-failure detection, the data are encoded through $\mathcal P$3 into a depth-limited Heisenberg-inspired ansatz and then projected onto single-qubit Pauli coefficients (Bowden et al., 22 Jan 2026).

4. Resource scaling, measurements, and software

A central claim of PQFMs is that they reduce the quantum bottleneck by measuring only a controlled observable family rather than estimating full state overlaps. In the Hamiltonian PQFM literature, the quantum stage scales linearly with the number of samples for a fixed number of observables, in contrast with the quadratic sample cost of fidelity-based kernels. In the active-subspace setting, the rank and sample-regularized dimension are controlled directly by the number of projected observables $\mathcal P$4, so the dimension blow-up of naive global kernels is avoided (Carmo et al., 11 Jun 2026).

Measurement design is therefore decisive. In the IoT projected-kernel study, local measurements with $\mathcal P$5 on six qubits produced 18 real features, nearest-neighbor two-qubit measurements with $\mathcal P$6 also produced 18, and the union produced 36. The same study emphasizes both the benefit and the cost of projection: Gram-matrix construction and SVM solving remain purely classical once measurements are in hand, but each kernel entry requires $\mathcal P$7 expectation estimates, hyperparameters proliferate, and real-device noise can break the PSD property of the Gram matrix. Shot-noise experiments showed convergence of accuracy with as few as approximately 1500 shots and good performance even at $\mathcal P$8 shots (d'Amore et al., 20 May 2025).

Hamiltonian PQFMs are explicitly engineered for shallow hardware execution. Single-step Trotter circuits use one- and two-qubit gates only; depth scales as $\mathcal P$9 rather than $$\rho(x)_M=\operatorname{Tr}_{\neg M}\!\left[|\psi(x)\rangle\langle\psi(x)|\right],$$0, and depending on the active axes one may require up to $$\rho(x)_M=\operatorname{Tr}_{\neg M}\!\left[|\psi(x)\rangle\langle\psi(x)|\right],$$1 more measurement settings. The biomedical benchmarking study reports that shot budgets of $$\rho(x)_M=\operatorname{Tr}_{\neg M}\!\left[|\psi(x)\rangle\langle\psi(x)|\right],$$2K suffice for stable expectation-value estimates, that the native coupling map and error rates can be respected during slot assignment, and that sample shuffling can be used for drift control (Carmo et al., 11 Jun 2026).

Implementation support has begun to consolidate. The publicly available Python library pqfmlib provides high-level QFM classes for CD-Ising, Heisenberg, and generalized XYZProjectiveQFM, hardware-aware feature-slot assignment utilities, automated extraction of one- and two-local expectation values, and nested cross-validation wrappers for benchmarking (Carmo et al., 11 Jun 2026). In industrial credit-default experiments, the PQF transformer also incorporated TREX readout-error symmetrization; after the feature map circuit, each qubit was measured in the $$\rho(x)_M=\operatorname{Tr}_{\neg M}\!\left[|\psi(x)\rangle\langle\psi(x)|\right],$$3, $$\rho(x)_M=\operatorname{Tr}_{\neg M}\!\left[|\psi(x)\rangle\langle\psi(x)|\right],$$4, and $$\rho(x)_M=\operatorname{Tr}_{\neg M}\!\left[|\psi(x)\rangle\langle\psi(x)|\right],$$5 bases, requiring three measurement sets and no ancilla or explicit qubit tracing (Ferenczi et al., 1 Oct 2025).

5. Benchmark domains and empirical findings

On four biomedical classification datasets—Molecular Toxicity, Breast Cancer Radiology I-SPY1, Wisconsin Diagnostic Breast Cancer, and Cleveland Heart Disease—the generalized Hamiltonian family was benchmarked against reference PQFMs under nested stratified 5-fold$$\rho(x)_M=\operatorname{Tr}_{\neg M}\!\left[|\psi(x)\rangle\langle\psi(x)|\right],$$610 repeats outer cross-validation, with one-sided Wilcoxon tests on fold-wise paired differences. Quantum features were generated using IBM quantum processors with up to 156 qubits and statevector simulations. The reported pattern was that generalized XYZ PQFMs, especially cross-axis and full diagonal+cross in multi-axis mode, showed the most consistent statistically supported AUC and recall gains over matched classical baselines across qubit budgets; one example given is a $$\rho(x)_M=\operatorname{Tr}_{\neg M}\!\left[|\psi(x)\rangle\langle\psi(x)|\right],$$7 AUC gain on 156 qubits for Molecular Toxicity (Carmo et al., 11 Jun 2026).

For IoT occupancy classification, all models were tuned by 10-fold cross-validation over $$\rho(x)_M=\operatorname{Tr}_{\neg M}\!\left[|\psi(x)\rangle\langle\psi(x)|\right],$$8 grids. The classical RBF SVM achieved $$\rho(x)_M=\operatorname{Tr}_{\neg M}\!\left[|\psi(x)\rangle\langle\psi(x)|\right],$$9 accuracy. Standard quantum kernels ranged from $$\phi_{\mathrm{proj}}(x)_{[(M,O)]}=\operatorname{Tr}[O\,\rho(x)_M], \qquad O\in\mathcal P.$$0 for the ZZ feature map to approximately $$\phi_{\mathrm{proj}}(x)_{[(M,O)]}=\operatorname{Tr}[O\,\rho(x)_M], \qquad O\in\mathcal P.$$1 for IQP/Trotterized embeddings. Projected quantum kernels matched or slightly exceeded the classical baseline depending on measurement choice: for $$\phi_{\mathrm{proj}}(x)_{[(M,O)]}=\operatorname{Tr}[O\,\rho(x)_M], \qquad O\in\mathcal P.$$2 and $$\phi_{\mathrm{proj}}(x)_{[(M,O)]}=\operatorname{Tr}[O\,\rho(x)_M], \qquad O\in\mathcal P.$$3, the best result was $$\phi_{\mathrm{proj}}(x)_{[(M,O)]}=\operatorname{Tr}[O\,\rho(x)_M], \qquad O\in\mathcal P.$$4 with the 3D+CNOT circuit, but the gain over the classical RBF baseline was reported as not statistically significant ($$\phi_{\mathrm{proj}}(x)_{[(M,O)]}=\operatorname{Tr}[O\,\rho(x)_M], \qquad O\in\mathcal P.$$5). The same study found that the simplest RotX and 3D embeddings generally outperformed deeper, heavily entangled maps such as ZZFeatureMap and Trotterized evolution (d'Amore et al., 20 May 2025).

In credit-default prediction, projected quantum features derived from a Heisenberg map were used as inputs to XGBoost and then combined with a purely classical XGBoost through mean and logistic-regression meta-ensembles. On balanced 1,000-sample subsamples, simulator accuracy rose to approximately $$\phi_{\mathrm{proj}}(x)_{[(M,O)]}=\operatorname{Tr}[O\,\rho(x)_M], \qquad O\in\mathcal P.$$6 at $$\phi_{\mathrm{proj}}(x)_{[(M,O)]}=\operatorname{Tr}[O\,\rho(x)_M], \qquad O\in\mathcal P.$$7 qubits, compared with approximately $$\phi_{\mathrm{proj}}(x)_{[(M,O)]}=\operatorname{Tr}[O\,\rho(x)_M], \qquad O\in\mathcal P.$$8 classically. On hardware, an example run on ibm_marrakesh with 101 qubits, 174 features, and 500 samples gave $$\phi_{\mathrm{proj}}(x)_{[(M,O)]}=\operatorname{Tr}[O\,\rho(x)_M], \qquad O\in\mathcal P.$$9 quantum accuracy versus $|S(n)_m|\cdot|\mathcal P|$0 classically, with noise-mitigated PQF circuits retaining no more than about $|S(n)_m|\cdot|\mathcal P|$1 absolute loss relative to simulator performance. On an unbalanced 10,000-sample setting, the reported improvement in Composite Default Risk over the purely classical model was $|S(n)_m|\cdot|\mathcal P|$2 for the mean ensemble in MPS simulation and $|S(n)_m|\cdot|\mathcal P|$3 for the logistic-regression ensemble on hardware (Ferenczi et al., 1 Oct 2025).

In machine-failure detection, projected quantum models were coupled to a statistical change-point detection pipeline and executed on IBM’s 133-qubit Heron quantum processor. The reported qualitative effect was that the projected quantum features sharpened change-point signatures in noisy multivariate time series; quantitatively, the quantum-transformed uLSIF outperformed classical uLSIF in 6 of 7 industrial sequences by AUC and reduced false alerts in real IoT vibration data (Bowden et al., 22 Jan 2026).

An analytically transparent example is provided by active quantum subspaces. In a 64-qubit Clifford family, one projected quantum feature

$|S(n)_m|\cdot|\mathcal P|$4

compresses an eight-way classical interaction into one scalar. On a synthetic contextual classification task with labels $|S(n)_m|\cdot|\mathcal P|$5, augmenting a simple linear model with this single quantum feature achieved near-perfect accuracy with $|S(n)_m|\cdot|\mathcal P|$6 samples, whereas purely classical RBF-kernel or degree-8 polynomial-kernel baselines were reported not to match this efficiency without expanding to thousands of features (Bang et al., 30 May 2026).

6. Limitations, interpretive cautions, and open directions

The PQFM literature is explicit that the present evidence is not a proof of quantum advantage. In the generalized Hamiltonian study, predictive gains depended on the dataset, encoding strategy, measured observables, and hardware conditions, and the authors described their results as support for a promising route toward near-term quantum utility rather than a definitive separation (Carmo et al., 11 Jun 2026).

Several technical limitations recur. The IoT projected-kernel study identified measurement overhead, hyperparameter explosion, and sensitivity to NISQ noise as practical obstacles, and it reported that entangled maps such as ZZFeatureMap and Trotterized circuits underperformed simple local rotations on that task. In the same work, real-device decoherence and gate errors were noted as a source of bias in $|S(n)_m|\cdot|\mathcal P|$7 estimates and as a possible cause of Gram matrices losing PSD-ness, with classical shadows and zero-noise extrapolation mentioned as possible but costly mitigation paths (d'Amore et al., 20 May 2025).

Projection itself is a deliberate truncation. In machine-failure detection, the reduction from the full $|S(n)_m|\cdot|\mathcal P|$8 degrees of freedom of $|S(n)_m|\cdot|\mathcal P|$9 to $O_{3j}=X_j$0 real numbers was presented as the source of tractability, but the same paper emphasized that truncating to 1-RDMs discards entanglement correlations and that some tasks may require 2-RDM or higher-order marginals. The proposed extensions were correspondingly concrete: selective higher-order marginals, combination with classical deep networks or kernels, automated ansatz search, and Fourier or frequency-spectrum analysis of the feature map (Bowden et al., 22 Jan 2026).

A further caution concerns simulability and encoding scope. One-dimensional Heisenberg PQFMs can be classically simulable by matrix-product-state methods, so not every projected construction is intrinsically hard for classical computation (Carmo et al., 11 Jun 2026). Conversely, active quantum subspace methods argue that full quantum encoding of the entire input is not necessary: only an information-bearing subset need be coherently lifted, while the remaining variables stay classical. This suggests a scaling strategy in which the projected quantum sector is used only where it contributes directions outside the classical feature span and correlated with the classical residual (Bang et al., 30 May 2026).